# Correlations of multiplicative functions and applications

Research paper by **Oleksiy Klurman**

Indexed on: **28 Mar '16**Published on: **28 Mar '16**Published in: **Mathematics - Number Theory**

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#### Abstract

We give an asymptotic formula for correlations \[ \sum_{n\le
x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are
bounded "pretentious" multiplicative functions, under certain natural
hypotheses. We then deduce several desirable consequences:\ First, we
characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with
bounded partial sums. This answers a question of Erd\H{o}s from $1957$ in the
form conjectured by Tao. Second, we show that if the average of the first
divided difference of multiplicative function is zero, then either $f(n)=n^s$
for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an
old conjecture of K\'atai. Third, we apply our theorem to count the number of
representations of $n=a+b$ where $a,b$ belong to some multiplicative subsets of
$\mathbb{N}.$ This gives a new "circle method-free" proof of the result of
Br\"udern.